
where \(x_{i}\)  (\(i=1,2,3\) ) refers to the undeformed configuration (material frame) and \(y_{i}\)  to the deformed coordinates (spatial frame).
We define the deformation gradient tensor and its inverse as [1]}
\(\mathsf {F}_{ij} = \frac{\partial {y_{i}}}{\partial {x_{j}}} = \frac{\partial {u_i}}{\partial {x_{j}}} + \delta _{ij}\quad \mathrm {and} \quad \mathsf {F}_{ij}^{-1}= \frac{\partial {x_{i}}}{\partial {y_{j}}}\:.\) 
